The case of non-relativistic particles is also of real practical interest for the applications of effective field theories in other branches of physics. This is so, even though one might think that an effective theory should contain only particles which are very light. Non-relativistic particles can nevertheless arise in practice within an effective field theory, even particles having masses which are large compared to those of the particles which were integrated out to produce the effective field theory in the first place. Such massive particles can appear consistently in a low-energy theory provided they are stable (or extremely long-lived), and so cannot decay and release enough energy to invalidate the low-energy approximation. Some well-known examples of this include the low-energy nuclear interactions of nucleons (as described within chiral perturbation theory [152, 153, 100, 109] ), the interactions of heavy fermions like the $b$  and $t$  quark (as described by heavy-quark effective theory (HQET) [90, 91, 108] ), and the interactions of electrons and nuclei in atomic physics (as described by non-relativistic quantum electrodynamics (NRQED) [34, 111, 103, 104, 127, 110] ).

The key to understanding the effective field theory for very massive, stable particles at low energies lies in the recognition that their anti-particles need not be included since they would have already been integrated out to obtain the effective field theory of interest. As a result heavy-particle lines within the Feynman graphs of the effective theory only directly connect external lines, and never arise as closed loops.

The most direct approach to estimating the size of quantum corrections in this case is to power-count as before, subject to the restriction that graphs including internal closed loops of heavy particles are to be excluded. Donoghue and Torma [60] have performed such a power-counting analysis along these lines, and shows that quantum effects remain suppressed by powers of light -particle energies (or small momentum transfers) divided by $M_p$  through the first few nontrivial orders of perturbation theory. Although heavy-particle momenta can be large, $p\gg M_p$  , they only arise in physical quantities through the small relativistic parameter $p/m_{\phi }\sim v$  rather than through $p/M_p$  , extending the suppression of quantum effects obtained earlier to non-relativistic problems.

Unfortunately, if a calculation is performed within a covariant gauge, individual Feynman graphs can depend on large powers like $m_{\phi }/M_p$  , even though these all cancel in physical amplitudes.

For this reason an all-orders inductive proof of the above power-counting remains elusive. As Donoghue and Torma [60] also make clear, progress towards such an all-orders power-counting result is likely to be easiest within a physical, non-covariant gauge, since such a gauge allows powers of small quantities like $v$  to be most easily followed.

Non-relativistic effective field theory

If experience with electromagnetism is any guide, effective field theory techniques are also likely to be the most efficient way to systematically keep track of both the expansion in inverse powers of both heavy masses, $1/M_p$  and $1/m_{\phi }$  – particularly for bound orbits. Relative to the theories considered to this point, the effective field theory of interest has two unusual properties. First, since it involves very slowly-moving particles, Lorentz invariance is not simply realized on the corresponding heavy-particle fields. Second, since the effective theory does not contain antiparticles for the heavy particles, the heavy fields which describe them contain only positive-frequency parts.

To illustrate how these features arise, we briefly sketch how such a non-relativistic effective theory arises once the antiparticles corresponding to a heavy stable particle are integrated out. We do so using a toy model of a single massive scalar field, and we work in position space to facilitate the identification of the couplings to gravitational fields.

Consider, then, a complex massive scalar field (we take a complex field to ensure low-energy conservation of heavy-particle number) having action

$$\begin{array}{c}-\frac{\mathcal{ℒ}}{\sqrt{-g}}=g^{\mu \nu }{\partial }_{\mu }{\phi }^{*}{\partial }_{\mu }\phi +m_{\phi }^2{\phi }^{*}\phi ,\end{array}$$

(36)

for which the conserved current for heavy-particle number is

$$\begin{array}{c}{J}^{\mu }=-i{g}^{\mu \nu }({\phi }^{*}{\partial }_{\nu }\phi -{\partial }_{\nu }{\phi }^{*}\phi ).\end{array}$$

(37)

Our interest is in exhibiting the leading couplings of this field to gravity, organized in inverse powers of $m_{\phi }$  . We imagine therefore a family of observers relative to whom the heavy particles move non-relativistically, and whose foliation of spacetime allows the metric to be written as

$$\begin{array}{c}d{s}^2=-(1+2\phi )d{t}^2+2N_{i}dtd{x}^i+{\gamma }_{ij}d{x}^{i}d{x}^j,\end{array}$$

(38)

where $i=1,2,3$  labels coordinates along the spatial slices which these observers define.

When treating non-relativistic particles it is convenient to remove the rest mass of the heavy particle from the energy, since (by assumption) this energy is not available to other particles in the low-energy theory. For the observers just described this can be done by extracting a time-dependent phase from the heavy-particle field according to $\phi \left(x\right)=F{e}^{-i{m}_{\phi }t}\chi \left(x\right)$  . $F=(2m_{\phi }{)}^{-1/2}$  is chosen for later convenience, to ensure a conventional normalization for the field $\chi$  . With this choice we have ${\partial }_t\phi =F({\partial }_t-i{m}_{\phi })\chi$  , and the extra $m_{\phi }$  -dependence introduced this way has the effect of making the large $m_{\phi }$  limit of the positive-frequency part of a relativistic action easier to follow.

With these variables the action for the scalar field becomes

$$\begin{array}{c}-\frac{\mathcal{ℒ}}{\sqrt{-g}}=\frac{m_{\phi }}{2}\left(g^{tt}+1\right){\chi }^{*}\chi +\frac{i}{2}{g}^{t\mu }\left({\chi }^{*}{\partial }_{\mu }\chi -{\partial }_{\mu }{\chi }^{*}\chi \right)+\frac{1}{2m_{\phi }}{g}^{\mu \nu }{\partial }_{\mu }{\chi }^{*}{\partial }_{\nu }\chi ,\end{array}$$

(39)

and the conserved current for heavy-particle number becomes

$$\begin{array}{c}{J}^{\mu }=-g^{\mu t}{\chi }^{*}\chi -\frac{i}{2m_{\phi }}{g}^{\mu \nu }({\chi }^{*}{\partial }_{\nu }\chi -{\partial }_{\nu }{\chi }^{*}\chi ).\end{array}$$

(40)

Here $g^{tt}=-1/D$  , $g^{ti}=N^i/D$  , and $g^{ij}={\gamma }^{ij}-N^i{N}^j/D$  , with $N^i={\gamma }^{ij}{N}_j$  and $D=1+2\phi +{\gamma }^{ij}{N}_i{N}_j$  .

Notice that for Minkowski space, where $g^{\mu \nu }={\eta }^{\mu \nu }=diag(-,+,+,+)$  , the first term in $\mathcal{ℒ}$  vanishes, leaving a result which is finite in the $m_{\phi }\to \infty$  limit. Furthermore – keeping in mind that the leading time and space derivatives are of the same order of magnitude ( ${\partial }_t\sim {\nabla }^2/m_{\phi }$  ) – the leading large $m_{\phi }$  part of $\mathcal{ℒ}$  is equivalent to the usual non-relativistic Schrödinger Lagrangian density, ${\mathcal{ℒ}}_{sch}={\chi }^{*}\left[i{\partial }_t+{\nabla }^2/\left(2m_{\phi }\right)\right]\chi$  . In the same limit the density of $\chi$  particles also takes the standard Schrödinger form $\rho =J^t={\chi }^{*}\chi +\mathcal{O}(1/m_{\phi })$  .

The next step consists of integrating out the anti-particles, which (by assumption) cannot be produced by the low-energy physics of interest. In principle, this can be done by splitting the relativistic field $\chi$  into its positiveand negative-frequency parts ${\chi }_{(\pm )}$  , and performing the functional integral over the negative-frequency part ${\chi }_{(-)}$  . (To leading order this often simply corresponds to setting the negative-frequency part to zero.) Once this has been done the fields describing the heavy particles have the non-relativistic expansion

$$\begin{array}{c}{\chi }_{(+)}\left(x\right)=\int d^{3}p{a}_p{u}_p^{(+)}\left(x\right),\end{array}$$

(41)

with no anti-particle term involving $a_p^{*}$  . It is this step which ensures the absence of virtual heavy-particle loops in the graphical expansion of amplitudes in the low-energy effective theory.

Writing the heavy-particle action in this way extends the standard parameterized post-Newtonian (PPN) expansion [68, 66, 67, 146] to the effective quantum theory, and so forms the natural setting for an all-orders power-counting analysis which keeps track of both quantum and relativistic effects.

For instance, for weak gravitational fields having $\phi \sim N^2\sim {\gamma }_{ij}-{\delta }_{ij}\ll 1$  , the leading gravitational coupling for large $m_{\phi }$  may be read off from Equation ( 39 ) to be

$$\begin{array}{c}{\mathcal{ℒ}}_0\approx -\frac{m_{\phi }}{2}\sqrt{-g}\left(g^{tt}+1\right){\chi }^{*}\chi \approx -m_{\phi }\left(\phi +\frac{N^2}{2}\right){\chi }^{*}\chi ,\end{array}$$

(42)

which for $N_i=0$  reproduces the usual Newtonian coupling of the potential $\phi$  to the non-relativistic mass distribution. For several $\chi$  particles prepared in position eigenstates we are led in this way to considering the gravitational field of a collection of classical point sources.

The real power of the effective theory lies in identifying the subdominant contributions in powers of $1/m_{\phi }$  , however, and the above discussion shows that different components of the metric couple to matter at different orders in this small quantity. Once $\phi$  is shifted by the static non-relativistic Newtonian potential, however, the remaining contributions are seen to couple with a strength which is suppressed by negative powers of $m_{\phi }$  , rather than positive powers.

A full power-counting analysis using such an effective theory, along the lines of the analogous electromagnetic problems [34, 111, 103, 104, 127, 110] , would be very instructive.

3.3 Effective field theory in curved space

All of the explicit calculations of the previous sections are performed for weak gravitational fields, which are well described as perturbations about flat space. This has the great virtue of being sufficiently simple to make explicit calculations possible, including the widespread use of momentum-space techniques. Much less is known in detail about effective field theory in more general curved spaces, although its validity is implicitly assumed by the many extant calculation of quantum effects in curved space [19] , including the famous calculation of Hawking radiation [87, 88] by black holes. This section provides a brief sketch of some effective-field theory issues which arise in curved-space applications.

For quantum systems in curved space we are still interested in the gravitational action, Equation ( 28 ), possibly supplemented by a matter action such as that of Equation ( 36 ). As before, quantization is performed by splitting the metric into a classical background ${\hat{g}}_{\mu \nu }$  and a quantum fluctuation $h_{\mu \nu }$  according to $g_{\mu \nu }={\hat{g}}_{\mu \nu }+h_{\mu \nu }$  . A similar expansion may be required for the matter fields $\phi =\hat{\phi }+\delta \phi$  , if these acquire vacuum expectation values $\hat{\phi }$  .

The main difference from previous sections is that ${\hat{g}}_{\mu \nu }$  is not assumed to be the Minkowski metric. Typically we imagine the spacetime may be foliated by a set of observers, as in Equation ( 38 ),

$$\begin{array}{c}d{s}^2=-(1+2\phi )d{t}^2+2N_{i}dtd{x}^i+{\gamma }_{ij}d{x}^{i}d{x}^j.\end{array}$$

(43)

and we imagine that the slices of constant $t$  are Cauchy surfaces, in the sense that the future evolution of the fields is uniquely specified by initial data on a constant- $t$  slice. The validity of this initial-value problem may also require boundary conditions for the fluctuations $h_{\mu \nu }$  such as that they vanish at spatial infinity.

3.3.1 When should effective Lagrangians work?

Many of the general issues which arise in this problem are similar to those which arise outside of gravitational physics when quantum fields are considered in the presence of background classical scalar or electromagnetic fields. In particular, there is a qualitative difference between the cases where the background fields are time-independent or time-dependent. The purpose of this section is to argue that we expect an effective field theory to work provided that the background fields vary sufficiently slowly with respect to time, an argument which in its relativistic context is called the `nice slice' criterion [130] .

Time-independent backgrounds

In flat space, background fields which are time-translation invariant allow the construction of a conserved energy $H$  , which evolves forward the fields from one $t$  slice to another. If $H$  is bounded below ^$\text{3}$  , then a stable ground state for the quantum fields (the `vacuum') typically exists, about which small fluctuations can be explored perturbatively. Since energy can be defined and is conserved, it can be used as a criterion for defining an effective theory which distinguishes between states which are `low-energy' or `high-energy' as measured using $H$  . Once this is possible, a low-energy effective theory may be defined, and we expect the general uncertainty-principle arguments given earlier to ensure that it is local in time.

A similar statement holds for background gravitational fields, with a conserved Hamiltonian following from the existence of a time-like Killing vector field $K^{\mu }$  for the background metric ${\hat{g}}_{\mu \nu }$  ^$\text{4}$  . For matter fields $H$  is defined in terms of $K^{\mu }$  and the stress tensor $T_{\mu \nu }$  by the integral

$$\begin{array}{c}{H}_{mat}={\int }_{t}d{\Sigma }_{\mu }{K}^{\nu }{T}_{\nu }^{\mu }.\end{array}$$

(44)

Here the integral is over any spacelike slice whose measure is $d{\Sigma }_{\mu }$  . Given appropriate boundary conditions this definition can also be extended to the fluctuations $h_{\mu \nu }$  of the gravitational field [10, 11, 12, 6, 16, 8, 7, 13, 14, 15, 1] .

If the timelike Killing vector $K^{\mu }$  is hypersurface orthogonal, then the spacetime is called static, and it is possible to adapt our coordinates so that $K^{\mu }{\partial }_{\mu }={\partial }_t$  and so the metric of Equation ( 43 ) specializes to $N_i=0$  , with $\phi$  and ${\gamma }_{ij}$  independent of $t$  . In this case we call the observers corresponding to these coordinates `Killing' observers. In order for these observers to describe physics everywhere, it is implicit that the timelike Killing vector $K^{\mu }$  be globally defined throughout the spacetime of interest.

If $H$  is defined and is bounded from below, its lowest eigenstate defines a stable vacuum and allows the creation of a Fock space of fluctuations about this vacuum. As was true for non-gravitational background fields, in such a case we might again expect to be able to define an effective theory, using $H$  to distinguish `low-energy' from `high-energy' fluctuations about any given vacuum.

It is often true that $K^{\mu }$  is not unique because there is more than one globally-defined timelike Killing vector in a given spacetime. For instance, this occurs in flat space where different inertial observers can be rotated, translated, or boosted relative to one another. In this case the Fock space of states to which each of the observers is led are typically all unitarily equivalent to one another, and so each observer has an equivalent description of the physics of the system.

An important complication to this picture arises when the timelike Killing vector field cannot be globally defined throughout all of the spacetime. In this case horizons exist, which divide the regions having timelike Killing vectors from those which do not. Often the Killing vector of interest becomes null at the boundaries of these regions. Examples of this type include the accelerated `Rindler' observers of flat space, as well as the static observers outside of a black hole. In this case it is impossible to foliate the entire spacetime using static slices which are adapted to the Killing observer, and the above construction must be reconsidered.

Putting the case of horizons aside for the moment, we expect that a sensible low-energy/high-energy split should be possible if the background spacetime is everywhere static, and if the conserved energy $H$  is bounded from below.

^$\text{3}$  Examples where $H$  is not bounded from below can arise, such as for charged particles in a sufficiently strong background electric field [133] . In such situations the runaway descent of the system to arbitrarily low energies is interpreted as being due to continual particle pair production by the background field.

^$\text{4}$  A Killing vector field satisfies the condition ${\hat{\nabla }}_{\mu }{\hat{K}}_{\nu }+{\hat{\nabla }}_{\nu }{\hat{K}}_{\mu }=0$  , which is the coordinate-invariant expression of the existence of a time-translation invariance of the background metric. The carets indicate that the derivatives are defined, and the indices are lowered by the background metric ${\hat{g}}_{\mu \nu }$  .

Time-dependent backgrounds

In non-gravitational problems, time-dependence of the background fields need not completely destroy the utility of a Hamiltonian or of a ground state, provided that this time dependence is sufficiently weak as to be treated adiabatically. In this case the lowest eigenstate of $H\left(t\right)$  for any given time defines both an approximate ground state and an energy in terms of which low-energy and high-energy states can be defined (such as by using an appropriate eigenvalue $\Lambda \left(t\right)$  of $H\left(t\right)$  ).

Once the system is partitioned in this way into low-energy and high-energy state, one can ask whether a purely low-energy description of time evolution is possible using only a low-energy, local effective Lagrangian. The main danger which arises with time-dependent backgrounds is that the time evolution of the system need not keep low-energy states at low energies, or high-energy states at high energies. There are several ways in which this might happen:

• ∙ The time dependence of the background can be rapid, for instance having Fourier components which are larger than the dividing line $\Lambda \left(t\right)$  between low-energy and high-energy states. If so, the background evolution is not sufficiently adiabatic to prevent the direct production of `heavy' particles, and an effective field theory need not exist. The criterion for this not to happen would be that the energies of any produced particles be smaller than the dividing eigenvalue $\Lambda \left(t\right)$  , which typically requires that $\Lambda \left(t\right)$  be much smaller than the background time dependence, for instance if $\Lambda \ll \dot{\Lambda }/\Lambda$  .
• ∙ Even for slowly-varying backgrounds there can be other dangers, such as the danger of level-crossing. For instance the dividing eigenvalue $\Lambda \left(t\right)$  may slowly evolve to smaller values and so eventually invalidate an expansion in inverse powers of $\Lambda \left(t\right)$  . In this case states which are prepared in the low-energy regime may evolve out of it, leading to a breakdown in the low-energy approximation. For instance, this might happen if $\Lambda \left(t\right)$  were chosen to be the mass of an initially heavy particle, which is integrated out to obtain the effective theory.
  If evolution of the background fields allows this mass to evolve to become too low, then eventually it becomes a bad approximation to have integrated it out.
• ∙ A related potential problem can arise if states whose energy is initially larger than $\Lambda \left(t\right)$  enter the low-energy theory by evolving into states having energy lower than $\Lambda \left(t\right)$  ^$\text{5}$  . This usually is only a problem for the effective-theory formulation if the states which enter in this way are not in their ground state when they do so, since then new physical excitations would arise at low energies. Conversely, they are not a problem for the effective field theory description so long as they enter in their ground states, as can usually be ensured if the background time evolution is adiabatic.

What emerges from this is that an effective field theory can make sense despite the presence of time-dependent backgrounds, provided one can focus on the evolution of low-energy states ( $E<\Lambda \left(t\right)$  ) without worrying about losing probability into high-energy states ( $E>\Lambda \left(t\right)$  ). This is usually ensured if the background time evolution is sufficiently adiabatic.

A similar story should also hold for background spacetimes which are not globally static, but for which a globally-defined timelike hypersurface-orthogonal vector field $V^{\mu }$  exists. For such a spacetime the observers for whom $V^{\mu }$  is tangent to world-lines can define a foliation of spacetime, as in Equation ( 43 ), but with the various metric components not being $t$  -independent. In this case the quantity defined by Equation ( 44 ) need not be conserved, $H=H\left(t\right)$  , for these observers. A low-energy effective theory should nonetheless be possible, provided $H\left(t\right)$  is bounded from below and is sufficiently slowly varying (in the senses described above). If such a foliation of spacetime exists, following [130] we call it a `nice slice'.

3.3.2 General power counting

Given an effective field theory, the next question is to analyze systematically how small energy ratios arise within perturbation theory. Since the key power-counting arguments of the previous sections were given in momentum space, a natural question is to ask how much of the previous discussion need apply to quantum fluctuations about more general curved spaces. In particular, does the argument that shows how large quantum effects are at arbitrary-loop order apply more generally to quantum field theory in curved space?

An estimate of higher-loop contributions performed in position space is required in order to properly apply the previous arguments to more general settings. Such a calculation is possible because the crucial part of the earlier estimate relied on an estimate for the high-energy dependence of a generic Feynman graph. This estimate was possible on dimensional grounds given the high-energy behaviour of the relevant vertices and propagators. The analogous computation in position space is also possible, where it instead relies on the operator product expansion [156, 41] . In position space one's interest is in how effective amplitudes behave in the short-distance regime, rather than the limit of high energy. But the short-distance limit of propagators and vertices are equally well-known, and resemble the short-distance limit which is obtained on flat space.

Consequently general statements can be made about the contributions to the low-energy effective theory.

Physically, the equivalence of the short-distance position-space and high-energy momentum-space estimates is expected because the high-energy contributions arise due to the propagation of modes having very small wavelength $\lambda$  . Provided this wavelength is very small compared with the local radius of curvature $r_c$  particle propagation should behave just as if it had taken place in flat space. One expects the most singular behaviour to be just as for flat space, with curvature effects appearing in subdominant corrections as powers of $\lambda /r_c$  .

Unfortunately, although the result is not in serious doubt, such a general position-space estimates for gravitational physics on curved space has not yet been done explicitly at arbitrary orders of perturbation theory. Partial results are known, however, including general calculations of the leading one-loop ultraviolet divergences in curved space [75, 36, 37, 115] .

3.3.3 Horizons and large redshifts

Among the most interesting applications of effective field theory ideas to curved space is the study of quantum effects near black holes and in the early universe. In particular, for massive black holes ( $M\gg M_p$  ) one expects semi-classical arguments to be valid since the curvature at the horizon is small and the interesting phenomena (like Hawking radiation) rely only on the existence of the horizon rather than on any properties of the spacetime near the internal curvature singularity [141] .

Although the power-counting near the horizon has not been done in the same detail as it has been for the asymptotic regions, semi-classical effective-field-theory arguments at the horizon are expected to be valid. Similar statements are also expected to be true for calculations of particle production in inflationary universes.

An objection has been raised to the validity of effective field theory arguments in both the black hole [139, 93, 94] and inflationary [95, 114, 23] contexts. For both of these cases the potential difficulty arises if one compares the energy of the modes as measured by different observers situated throughout the spacetime. For instance, a mode which emerges far from a black hole at late times with an energy (as seen by static and freely-falling observers) close to the Hawking temperature starts off having extremely high energies as seen by freely-falling observers very close to, but outside of, the black hole's event horizon just as it forms. The energy measured at infinity is much smaller because the state experiences an extremely large redshift as it climbs out of the black hole's gravitational well. The corresponding situation in inflation is the phenomenon in which modes get enormously redshifted (all the way from microscopic to cosmological scales) as the universe expands.

It has been argued that these effects prevent a consistent low-energy effective theory from being built in these situations, because very high-energy states are continuously turning up at later times at low energies. If so, this would seem to imply that a reliable calculation of phenomena like Hawking radiation (or inflationary fluctuations of the CMB) necessarily require an understanding of very high energy physics. Since we do not know what this very high energy physics is, this is another way of saying that these predictions are theoretically unreliable, since uncontrolled theoretical errors potentially contribute with the same size as the predicted effect.

The remainder of this section argues that although the concerns raised are legitimate, they are special cases of the general conditions mentioned earlier which govern the applicability of effective field theory ideas to time-dependent backgrounds also in non-gravitational settings. As such one expects to find robustness against adiabatic physics at high energies, and sensitivity to non-adiabatic effects. (This is borne out by the explicit calculations to date.) Given a concrete theory of what the high-energy physics is, one can then ask into which category it falls, and so better quantify the theoretical error.

Black holes

For black holes the potential high-energy difficulty can be traced to the fact that the energy of an freely-falling object as seen by static observers becomes arbitrarily large as one approaches the event horizon. This is why the escaping modes have such high energies as seen by static observers near the horizon. This raises two separate issues for effective field theories, which are worth separating: the issue of the relevance of static observers measuring very large energies for freely-falling objects, and the issue of high-energy states descending into the low-energy theory as time progresses. Each of these is now discussed in turn.

The fact that freely-falling observers measure different energies for outgoing particles, depending on their distance from the horizon, underlines that there is a certain amount of frame dependence in any effective-theory description, even in flat space. This is so because energy is used as the criterion for deciding which states fall into the effective theory and which do not, yet any nominally low-energy particle has a large energy as seen by a sufficiently boosted observer. In practice this is not a problem, because the validity of the effective theory description only requires the existence of low-energy observers, not that all observers be at low energy. What is important is that the physically relevant energies for the process of interest – for instance, the centre-of-mass energies in a scattering event – are small in order for this process to be describable using a low-energy theory.

Once this is true, invariant quantities like cross sections take a simple low-energy form when expressed in terms of physical kinematic variables, regardless of the energies which the particles involved have as measured by observers who are highly boosted compared to the centre-of-mass frame.

Flat-space experience therefore suggests that there need not be a problem associated with escaping modes having large energies as seen by freely-falling observers. This only indicates that the use of some observers near the horizon may be problematic. So long as the physics involved does not rely crucially on these observers, it may in any case allow an effective-theory description.

This is essentially the point of view put forward in [93, 94] and [130]  ^$\text{6}$  , where it is argued that the robustness of the Hawking radiation to high-energy physics is most simply understood if one is careful to foliate the spacetime using slices which are chosen to be `nice slices' (in the sense described above), which cut through the horizon in such a way as to encounter only small curvatures and adiabatic time variation. Since such slices exist, a low-energy theory may be set up in terms of the slowly-varying $H\left(t\right)$  which these slices define. A great many calculations using nice slices have been done, including for example [44, 47, 125, 117, 118] .

Of course, calculations need not explicitly use the nice slices in order to profit from their existence. In the same way that dimensional regularization can be more useful in practice for calculations in effective field theories, despite its inclusion of arbitrarily high energy modes, the sensitivity of Hawking radiation to high energies can be investigated using a convenient covariant regularization. This is because if nice slices exist, covariant calculations must reproduce the insensitivity to high energies which they guarantee. This is borne out by explicit calculations of the sensitivity of the Hawking radiation to high energies [86] using a simple covariant regularization.

The most important manner in which high-energy states can influence the Hawking radiation has been identified from non-covariant studies, such as those which model the high-energy physics as non-Lorentz invariant dispersion relations for otherwise free particles [140, 24, 42] . (See [95] for a review, with references, of these calculations.) These identify the second pertinent issue mentioned above: the descent of higher-energy states into the low-energy theory. In these calculations high-energy modes cross into the low-energy theory because of their redshift as they climb out of the gravitational potential well of the black hole. The usual expression for the Hawking radiation follows provided that these modes enter the low-energy theory near the black hole horizon in their adiabatic ground state (a result which can also be seen in covariant approaches, where it can be shown that the Hawking radiation depends only on the form of the singularity of the propagator near the light cone [71] ). If these modes do not start off in their ground state, then they potentially cause observable changes to the Hawking radiation.

The condition that high-energy modes enter the low-energy theory in their ground state is reminiscent of the same condition which was encountered in previous sections as a general pre-condition for the validity of a low-energy effective description when there are time-dependent backgrounds (including, for example, the descent of Landau levels in a decreasing magnetic field). In non-gravitational contexts it is automatically satisfied if the background evolution is adiabatic, and this can also be expected to be true in the gravitational case. Of course, this expectation cannot be checked explicitly unless the theory for the relevant high-energy physics is specified, but it is borne out by all of the existing calculations. To the extent that high-energy modes do not arise in their adiabatic vacua, their effects might be observable in the Hawking radiation as well as in possibly many other observables which would otherwise be expected to be insensitive to high-energy physics.

Clearly this is good news, since it tells us that we can believe that generic quantum effects do not ruin the classical calculations using general relativity, which tell us that black holes exist. Nor do they ruin the semiclassical calculations which lead to effects like the Hawking radiation [87, 88] in the vicinity of black holes – provided that the black hole mass is much larger than $M_p$  (which we shall see is required if quantum effects are to remains small at the event horizon). On the other hand, it means that we cannot predict the final stages of black hole evaporation, since these inevitably lead to small black hole masses, where the semiclassical approximation breaks down.

^$\text{6}$  These authors have slightly different spins on the more philosophical question of whether trans-Planckian physics is likely to be found to be non-adiabatic.

Inflation

Many of the issues concerning the validity of effective field theories which arise for the Hawking radiation also arise within inflationary cosmology, and have generated considerable discussion due to the recent advent of precise measurements of CMB temperature fluctuations. By analogy with the black hole case, it has been proposed [95, 114, 23] that very-high-energy physics may not decouple from inflationary predictions due to the exponential expansion of space. If so, there might be detectable imprints on the observed temperature fluctuations in the cosmic microwave background [96] . Conversely, if high-energy effects do contaminate inflationary predictions for CMB fluctuations at an observable level, then inflationary predictions themselves must be recognized as containing an uncontrollable theoretical uncertainty. If so, their successful description of the observations cannot be deemed to be credible evidence of the existence of an earlier inflationary phase. There is clearly much at stake.

It is beyond the scope of the this article to summarize all of the intricacies associated with quantum field theory in de Sitter space, so we focus only on the parallels with the black hole situation. The bottom line for cosmology is similar to what was found for the Hawking radiation:

• ∙ Observable effects can be obtained from physics associated with energies $E$  much higher than the inflationary expansion rate $H$  , if the states associated with the heavy physics are chosen not to be in their adiabatic vacuum ^$\text{7}$  . Potentially observable effects have been obtained by explicit calculations which incorporate non-adiabatic physics of various types. For many of these the high-energy physics is modelled as a free particle having a non-Lorentz-invariant dispersion relation (for a review with references, see [22] ). However large Lorentz-violating interactions need not be required since similar effects are also obtained using more conventional inflationary field theories, for which background scalar fields are allowed to roll non-adiabatically [31] .
• ∙ If the states associated with high-energy physics are prepared in their adiabatic vacua then an effective field theory description applies. In this case most kinds of heavy physics decouple, and the vast majority of effects it can produce for the microwave background can be argued to be smaller than $\mathcal{O}(H^2/M^2)$  [97, 98] . Even in this case, however, larger contributions can be obtained using ordinary inflationary field-theory examples, where low-energy effects can instead be suppressed by powers of $m/M$  rather than $H/M$  , where $m$  is a light scale which need not be as small as $H$  [30] .

Again the final picture which emerges is encouraging. The criteria for validity of effective field theories appear to be the same for gravity as they are in non-gravitational situations. In particular, for a very broad class of high-energy physics effective field theory arguments apply, and so theoretical predictions for the fluctuations in the CMB are robust in the sense that they are insensitive to most of the details of this physics. But some kinds of high-energy effects can produce observable phenomena, and these should be searched for.

4 Explicit Quantum Calculations

Although the ideas presented in here have been around for more than twenty years [147] , explicit calculations based on them have only recently been made. Two of these are summarized in this section.

4.1 Non-relativistic point masses in three spatial dimensions

The first example to be considered consists of quantum corrections to the potential energy $V\left(r\right)$  of gravitational interaction for two large, slowly-moving point masses separated by a distance $r$  .

Working to leading order in source velocities $v$  , we expect the leading behavior for large source masses to be the Newtonian gravitational interaction of two classical, static point sources of energy:

$$\begin{array}{c}\rho (\mathbf{r})={\sum }_i{M}_i{\delta }^3(\mathbf{r}-{\mathbf{r}}_i).\end{array}$$

(45)

Our interest is in the quantum and relativistic corrections to this Newtonian limit, as described by the gravitational action, Equation ( 28 ), plus the appropriate source action (like, for instance, Equation ( 39 )). For point sources which are separated by a large distance $r$  we expect these corrections to be weak, and so they should be calculable in perturbation theory about flat space.

The strength of the gravitational interaction at large separation is controlled by two small dimensionless quantities, which suggest themselves on dimensional grounds. Temporarily re-instating factors of $ħ$  and $c$  , these small parameters are $Għ/r^2{c}^3$  and $G{M}_i/r{c}^2$  . Both tend to zero for large $r$  , and as we shall see, the first controls the size of quantum corrections and the second controls the size of relativistic corrections ^$\text{8}$  .

^$\text{8}$  The point of the non-relativistic power-counting of the previous section is to show that the third, large, $r$  -independent dimensionless quantity $G{M}_i{M}_j/ħc$  does not appear in the interaction energy.

4.1.1 Definition of the potential

Because there is some freedom of choice in the definition of an interaction potential in a relativistic field theory, we first pause to consider some of the definitions which have been considered.

Although more sophisticated possibilities are possible [122, 43] , for systems near the flat-space limit a natural definition of the interaction potential between slowly-moving point masses can be made in terms of their scattering amplitudes.

Consider, then, two particles which scatter non-relativistically, with each undergoing a momentum transfer, $\Delta {\mathbf{p}}_1=-\Delta {\mathbf{p}}_2\equiv \mathbf{q}$  , in the center-of-mass frame. The most direct definition of the interaction potential $V(\mathbf{r})$  of these two particles is to define its matrix elements within single-particle states to reproduce the full field-theoretical amplitude for this scattering. For instance, if the field-theoretic scattering matrix takes the form $\langle f\left|T\right|i\rangle =\left(2\pi {)}^4{\delta }^4\right(p_f-p_i)\mathcal{A}(q)$  , the potential $V$  would be defined by

$$\begin{array}{c}\langle f\left|T\right|i\rangle =2\pi \delta (E_f-E_i)\langle f\left|\tilde{V}\right(\mathbf{q}\left)\right|i\rangle .\end{array}$$

(46)

The position-space potential is then given by $V(\mathbf{r})=N(2\pi {)}^{-3}\int d^{3}q\tilde{V}(\mathbf{q})$  . The overall normalization $N$  depends on the conventions used for the normalization of the initial and final states, and is chosen to ensure the proper form for the Newtonian interaction.

Figure 2 : The 1-particle-reducible Feynman graphs relevant to the definition of the interaction potential. The blobs represent self-energy and vertex corrections.

Several other definitions for the interaction potential have also been considered by various workers, some of which we now briefly list.

One-particle-reducible amplitude

An alternative definition, followed in [56, 55, 5] , is to define the interaction potential in terms of the one-particle-reducible part of the amplitude ${\mathcal{A}}_{1PR}$  – see Figure  2 – as is commonly done for quantum electrodynamics and quantum chromodynamics. The logic of this choice is that because the graviton propagator varies as $1/q^2$  , for small $q^2$  graviton exchange dominates very-long-distance interactions. It has the disadvantage that the one-particle-reducible graphs are not observable in themselves, and so need not form a gauge-invariant subset. Nevertheless, the results obtained from this definition can be interpreted [59, 21] as giving the leading quantum corrections to the Schwarzschild, Kerr–Newman, and Reisner–Nordström metrics.

Vacuum polarization

Some early workers defined the interaction potential in terms of the purely vacuum polarization subset of the 1-particle-reducible graphs [62, 33, 79] . The motivation for such a choice is that these are the only graphs which would arise for a purely classical source, which macroscopic objects like planets or stars were expected to be. It is important to recognize that the power-counting arguments given earlier necessarily require the inclusion of vertex corrections at the same order in small quantities as the vacuum polarization graphs. The necessity for so doing shows that there is no limit in which a source for the gravitational field can be considered to be precisely classical.

This non-classicality arises because the gravitational field itself carries energy, and its quantum fluctuations do not decouple in the large-mass limit due to the growth which the gravitational coupling experiences in this limit.

4.1.2 Calculation of the interaction potential

We now describe the results of recent explicit calculations of the gravitational potential just defined. A number of these calculations have now been performed [81, 92, 82, 85, 101, 102, 89] , and it is the results of [56, 55, 21, 20] which are summarized here.

For any of these potentials, scattering at large distances ( $r\to \infty$  ) – i.e., large impact parameters – corresponds to small momentum transfers, ${\mathbf{q}}^2\to 0$  . Because corrections to the Newtonian limit involve the interchange of massless gravitons, in general scattering amplitudes are not analytic in this limit. In particular, in the present instance the small- ${\mathbf{q}}^2$  limit to the scattering amplitude turns out to behaves as

$$\begin{array}{c}\mathcal{A}(\mathbf{q})=\frac{k_2}{{\mathbf{q}}^2}+\frac{k_1}{\sqrt{{\mathbf{q}}^2}}+k_{0}log\left({\mathbf{q}}^2\right)+{\mathcal{A}}_{an}\left({\mathbf{q}}^2\right),\end{array}$$

(47)

where ${\mathcal{A}}_{an}=A_0+A_2{\mathbf{q}}^2+...$  is an analytic function of ${\mathbf{q}}^2$  near ${\mathbf{q}}^2=0$  .

In position space the first three terms of Equation ( 47 ) correspond to terms which fall off with $r$  like $k_2/r$  , $k_1/r^2$  , and $k_0/r^3$  , respectively. By contrast, the powers of ${\mathbf{q}}^2$  in ${\mathcal{A}}_{an}$  only contribute terms to $V(\mathbf{r})$  which are local, inasmuch as they are proportional to ${\delta }^3(\mathbf{r})$  or its derivatives. Since our interest is only in the long-distance interaction, the analytic contributions of ${\mathcal{A}}_{an}$  may be completely ignored in what follows.

The power-counting analysis described in earlier sections suggest that the leading corrections to the Newtonian result come either from (i) relativistic contributions coming from tree-level calculations within general relativity, (ii) one-loop corrections to the classical potential, again using only general relativity, or (iii) from tree-level contributions containing precisely one vertex from the curvature-squared terms of the effective theory, Equation ( 28 ). The interaction potential therefore has the form

$$\begin{array}{c}V\left(r\right)=V_{GR,cl}(M_p;r)+V_{GR,q}(M_p;r)+V_{cs}(a,b,M_p;r),\end{array}$$

(48)

respectively corresponding to the contributions of classical general relativity, the one-loop corrections within general relativity, and the classical curvature-squared contributions. The dependence of this latter term on the quantities $M_p$  , $a$  , and $b$  are written explicitly to emphasize which contributions depend on which parameters. We now describe the result which is obtained for each of these three types of contribution.

Curvature-squared terms

The simplest contribution to dispose of is that due to the curvature-squared terms ^$\text{9}$  . Because these terms are polynomials in momenta, they contribute only to the analytic part ${\mathcal{A}}_{an}$  of the scattering amplitudes, and so give only local contributions to the interaction potential which involve ${\delta }^3(\mathbf{r})$  or its derivatives. Their precise form is computed in [56, 55] , who find

$$\begin{array}{c}{V}_{cs}\left(r\right)=G{M}_1{M}_{2}B{\delta }^3(\mathbf{r}),\end{array}$$

(49)

with $B$  given in terms of the constants $a$  and $b$  of Equation ( 28 ) by $B=128{\pi }^{2}G(a+b)$  . Since they contribute only to ${\mathcal{A}}_{an}$  , we see that these contributions are necessarily irrelevant to the large-distance interaction potential.

It is instructive to think of this $\delta$  -function contribution due to curvature-squared terms in another way. To this end, consider the toy model of a massless scalar field coupled to a classical $\delta$  -function source, whose Lagrangian is

$$\begin{array}{c}-\mathcal{ℒ}=\frac{1}{2}(\partial \phi {)}^2+\frac{\kappa }{2}(\square \phi {)}^2.\end{array}$$

(50)

The higher-derivative term proportional to $\kappa$  in this model is the analogue of the curvature-squared gravitational interactions. The propagator for this theory satisfies the equation $(\square -\kappa {\square }^2)G_{\kappa }(x,y)={\delta }^4(x-y)$  , which becomes (to linear order in $\kappa$  )

$$\begin{array}{c}{G}_{\kappa }(x,y)\approx G_0(x,y)+\kappa \square G_0(x,y)=G_0(x,y)+\kappa {\delta }^4(x-y),\end{array}$$

(51)

where $G_0(x,y)$  is the usual propagator when $\kappa =0$  . We see the expected result that the leading contribution to $V\left(r\right)$  is purely local in position space (as might be expected for the low-energy implications of very-high-energy/very-short-range physics).

This way of thinking of things is useful because it illustrates an important conceptual issue for effective field theories. Normally one considers higher-derivative theories to be anathema since higher-derivative field equations generically have unstable runaway solutions, and the above calculation shows why these do not pose problems for the effective field theory. To see why this is so, it is useful to pause to review how the runaway solutions arise.

At the classical level, runaway modes are possible because of the additional initial data which higher-derivative equations require. The reason for their origin in the quantum theory is also easily seen using the toy theory defined by Equation ( 50 ), for which at face value the momentum-space scalar propagator would be

$$\begin{array}{c}-iG\left(p\right)\propto \frac{1}{p^2+\kappa p^4}=\frac{1}{p^2}-\frac{1}{p^2+{\kappa }^{-1}}.\end{array}$$

(52)

This shows how the higher-derivative term introduces a new pole into the propagator at $p^2=-{\kappa }^{-1}$  , but with a residue whose sign is unphysical (corresponding to a ghost mode with negative kinetic energy).

The reason these do not pose a problem for effective field theories is that all of the higher-derivative terms are required to be treated perturbatively, since these interactions are defined by reproducing the results of the underlying physics order-by-order in powers of inverse heavy masses $1/m$  . In the effective theory of Equation ( 50 ) the propagator ( 52 ) must be read as

$$\begin{array}{c}-iG\left(p\right)\propto \frac{1}{p^2}\left(1-\kappa p^2+...\right),\end{array}$$

(53)

since the Lagrangian itself is only accurate to leading order in $\kappa$  . The ghost pole does not arise perturbatively in $\kappa p^2$  , since its location is up at high energies, $p^2=-{\kappa }^{-1}$  . Simon [136] makes this general argument explicit for the specific case of higher-derivative gravity linearized about flat space.

^$\text{9}$  Notice that the curvature-squared terms can no longer be eliminated by performing field redefinitions once classical sources are included. Instead they can only be converted into the direct source-source interactions in which we are interested.

Classical general relativity

The leading contributions for large $r$  due to the relativistic corrections of general relativity have the large- $r$  form (with factors of $c$  restored)

$$\begin{array}{c}{V}_{GR,cl}\left(r\right)=-\frac{G{M}_1{M}_2}{r}\left[1+\lambda \frac{G(M_1+M_2)}{r{c}^2}+...\right],\end{array}$$

(54)

where $G=1/\left(8\pi M_p^2\right)$  is Newton's constant, $M_1$  and $M_2$  are the masses whose potential energy is of interest, and which are separated by the distance $r$  .

The square brackets, $\left[1+...\right]$  , in this expression represent the relativistic corrections to the Newtonian potential which already arise within classical general relativity, and $\lambda$  is a known constant whose value depends on the precise coordinate conditions used in the calculation. For example, using the potential defined by the 1-particle-reducible scattering amplitude gives ${\lambda }_{1PR}=-1$  [56, 55, 21] , corresponding to the classical result for the metric in harmonic gauge, for which the Schwarzschild metric takes the form

$$\begin{array}{c}{g}_{00}=-\frac{1-GM/r}{1+GM/r}=-1+2\left(\frac{GM}{r}\right)-2{\left(\frac{GM}{r}\right)}^2+....\end{array}$$

(55)

Alternatively, using the potential defined by the full scattering amplitude ${\mathcal{A}}_{tot}$  instead gives ${\lambda }_{tot}=+3$  [20] . It is natural that different values for $\lambda$  are obtained when different definitions for $V$  are used, since these different definitions contribute differently to physical observables (on which all calculations must agree).

There is another ambiguity in the definition of the potential [89] , which is related to the freedom to redefine the coordinate $r$  , according to $r\to r^{\prime }=r(1+aGM/r+...)$  . Of course, such a coordinate change should drop out of physical observables, but how this happens in this case involves a subtlety.

The main point is that the low-energy effective Lagrangian for the non-relativistic particles contains two terms of the same size at subleading order in the relativistic expansion, having the schematic form

$$\begin{array}{c}\Delta \mathcal{ℒ}=\lambda (G{M}^2/r)(GM/r)+{\lambda }^{\prime }(GM/r)\left(M{v}^2\right),\end{array}$$

(56)

where $M$  and $v$  are the mass and velocity of the non-relativistic particle of interest. The main point is that the constants $\lambda$  and ${\lambda }^{\prime }$  are redundant interactions in the sense defined earlier, inasmuch as all physical observables only depend on a single combination of these two constants. Observables only depend on one combination because the other combination can be removed by performing the coordinate transformation $r\to r(1+aGM/r+...)$  as above. From this we see that the coefficient $\lambda$  of $GM/r^2$  obtained for $V\left(r\right)$  can also differ from one another, provided that the coefficient ${\lambda }^{\prime }$  also differs in such a way as to give the same results for physical observables.

One-loop general relativity

The final term in $V\left(r\right)$  arises from the one-loop contribution as computed within general relativity, which is extracted by calculating the one-loop corrections to the scattering amplitude ${\mathcal{A}}_q$  . Although these corrections typically diverge in the ultraviolet, on general grounds such divergences contribute only polynomials in momenta, and so can contribute only to the non-relativistic amplitude's analytic part ${\mathcal{A}}_{an}\left({\mathbf{q}}^2\right)$  . Indeed, this is required for the one-loop divergences to be absorbed by renormalizing the effective couplings $a$  and $b$  of the higher-curvature terms of the gravitational action ( 28  ^$\text{10}$  .

It follows from this observation that to the extent that we focus on the long-distance interactions in $V\left(r\right)$  , to the order we are working these must be ultraviolet finite since they receive no contribution from the amplitude's analytic part. This means that the leading quantum implications for $V\left(r\right)$  are unambiguous predictions which are not complicated by the renormalization procedure.

Explicit calculation shows that the non-analytic part of the quantum corrections to scattering are proportional to $log{\mathbf{q}}^2$  , and so the leading one-loop quantum contribution to the interaction potential is (again re-instating powers of $ħ$  and $c$  )

$$\begin{array}{c}{V}_{GR,q}\left(r\right)=-\frac{G{M}_1{M}_2}{r}\left[1+\xi \frac{Għ}{r^2{c}^3}+...\right],\end{array}$$

(57)

where $\xi$  is a calculable number. If the potential is computed using only the one-particle-reducible scattering amplitude, the result for pure gravity is [21] :

$$\begin{array}{c}{\xi }_{1PR}=-\frac{167}{30\pi }.\end{array}$$

(58)

Notice that this corrects an error in the earlier result for the same quantity, given in [56, 55] . If, instead, the full amplitude ${\mathcal{A}}_{tot}$  is used to define the interaction potential, Bjerrum–Bohr et al. [20] find

$$\begin{array}{c}{\xi }_{tot}=+\frac{41}{10\pi }.\end{array}$$

(59)

It is argued in [20] that these one-loop results for $\xi$  do not suffer from ambiguity due to the freedom to perform redefinitions of the form $r\to r(1+a{G}^2/r^2+...)$  .

^$\text{10}$  The necessity for renormalizing $a$  and $b$  in addition to Newton's constant at one loop reflects the fact that general relativity is not renormalizable. Still higher-curvature terms would be required to absorb the divergences at two loops and beyond.

4.1.3 Implications

It is remarkable that the quantum corrections to the interaction potential can be so cleanly identified. In this section we summarize a few general inferences which follow from their size and dependence on physical parameters like mass and separation.

Conceptually, the main point is that the quantum effects are calculable, and in principle can be distinguished from purely classical corrections. For instance, the quantum contribution ( 57 ) can be distinguished from the classical relativistic corrections ( 54 ) because the quantum and the relativistic terms depend differently on $G$  and the masses $M_1$  and $M_2$  . In particular, relativistic corrections are controlled by the dimensionless quantity $G{M}_{tot}/r{c}^2$  , which is a measure of typical orbital velocities $v^2/c^2$  . The leading quantum corrections, on the other hand, are $M$  -independent and are controlled by the ratio ${\ell }_p^2/r^2$  , where ${\ell }_p=(Għ/c^3{)}^{1/2}\sim {10}^{-35}m$  is the Planck length.

Although the one-particle-reducible contributions need not be separately gauge-independent, Bjerrum–Borh [21] and Donoghue [59] argue that they may be usefully interpreted as defining long-distance quantum corrections to the metric external to various types of point sources. Besides obtaining corrections to the Schwarzschild metric in this way, they do the same for the Kerr–Newman and Reissner–Nordström metrics by incorporating spin and electric charge into the non-relativistic quantum source. Because the quantum corrections they find are source-independent, these authors suggest they be interpreted in terms of a running Newton's constant, according to

$$\begin{array}{c}G\left(r\right)=G\left[1-\frac{167}{30\pi }\left(\frac{G}{r^2}\right)+...\right].\end{array}$$

(60)

Numerically, the quantum corrections are so miniscule as to be unobservable within the solar system for the forseeable future. Table  1 evaluates their size using for definiteness a solar mass $M_{\odot }$  , and with $r$  chosen equal to the solar radius $R_{\odot }\sim {10}^{9}m$  , or the solar Schwarzschild radius $r_s=2G{M}_{\odot }/c^2\sim {10}^{3}m$  . Clearly the quantum-gravitational correction is numerically extremely small when evaluated for garden-variety gravitational fields in the solar system, and would remain so right down to the event horizon even if the sun were a black hole. At face value it is only for separations comparable to the Planck length that quantum gravity effects become important. To the extent that these estimates carry over to quantum effects right down to the event horizon on curved black hole geometries (more about this below) this makes quantum corrections irrelevant for physics outside of the event horizon, unless the black hole mass is as small as the Planck mass, $M_{hole}\sim M_p\sim {10}^{-5}g$  .

Table 1 : The generic size of relativistic and quantum corrections to the Sun's gravitational field.

$\frac{G{M}_{\odot }}{r{c}^2}$  $\frac{Għ}{r^2{c}^3}$  $r=R_{\odot }$  ${10}^{-6}$  ${10}^{-88}$  $r=2G{M}_{\odot }/c^2$  $0.5$  ${10}^{-76}$

Of course, the undetectability of these quantum corrections does not make them unimportant.

Rather, the above calculations underline the following three conclusions:

• ∙ One need not throw up one's hands when contemplating quantum gravity effects, because quantum corrections in gravity are often unambiguous and calculable.
• ∙ Although the small size of the above quantum corrections in the solar system mean that they are unlikely to be measured, they also show that the great experimental success of classical general relativity in the solar system should also be regarded as a triumph of quantum gravity! Classical calculations are not a poor substitute for some poorly-understood quantum theory, they are rather an extremely good approximation for which quantum corrections are exceedingly small.
• ∙ Despite the above two points, the mysteries of quantum gravity remain real and profound.
  But the above calculations show that these are high-energy (or short-distance) mysteries, and so point to cosmological singularities or primordial black holes as being the places to look for quantum gravitational effects.

4.2 Co-dimension two and cosmic strings

A second explicit calculation of quantum corrections within general relativity has been done for the gravitational field of a cosmic string, which for our purposes is a line distribution of mass characterized by a mass-per-unit-length $\rho$  . This system naturally suggests itself as a theoretical laboratory for computing quantum effects because its classical gravitational field is extremely simple.

The classical field due to a line distribution of mass is simple for the following reason. Because of the symmetry of the mass distribution, the calculation of the gravitational field it produces is effectively a $2+1$  dimensional problem. If the exterior to the mass distribution is empty, we seek there a solution to the vacuum Einstein equations $R_{\mu \nu }=0$  . But it is a theorem that in $2+1$  dimensions any geometry which is Ricci flat must also be Riemann flat: $R_{\mu \nu \lambda \rho }=0$  ! Superficially this appears to lead to the paradoxical conclusion that long, straight cosmic strings should not gravitate.

This conclusion is not quite correct, however. Although it is true that the vanishing of the Riemann tensor implies no tidal forces for test particles which pass by on the same side of the string, test particles are influenced to approach one another if they pass by on opposite sides of the string.

The reason for this may be seen by more closely examining the spacetime's geometry near the position of the cosmic string. The boundary conditions at this point require that spacetime there to resemble the tip of a cone, inasmuch as an infinitely thin cosmic string introduces a $\delta$  -function singularity into the curvature of spacetime. This implies that the flat geometry outside of the string behaves globally like a cone, corresponding to the removal of a defect angle, $\Delta \theta =8\pi G\rho$  radians, from the external geometry. This conical geometry for the external spacetime is what causes the focussing of trajectories of pairs of particles which pass by on either side of the string [49, 48] .

The above considerations show that the gravitational interaction of two cosmic strings furnishes an ideal theoretical laboratory for studying quantum gravity effects near flat space. Since the classical gravitational force of one string on the other vanishes classically, its leading contribution arises at the quantum level. Consider, for instance, the interaction energy per-unit-length $u_{int}$  of two straight parallel strings separated by a distance $a$  . This receives no contribution from the Einstein–Hilbert term of the effective action, for the reasons just described. Furthermore, just as for point gravitational sources, higher-curvature interactions only generate contact interactions, and so are also irrelevant for computing the strings' interactions at long range. The leading contribution therefore arises at the quantum level, and must be ultraviolet finite.

These expectations are borne out by explicit one-loop calculations, which have been computed [154] for the case of two strings having constant mass-per-unit-lengths ${\rho }_1$  and ${\rho }_2$  . The result obtained is (again temporarily restoring the explicit powers of $ħ$  and $c$  )

$$\begin{array}{c}{u}_{int}\left(a\right)=\left(\frac{24ħ}{5\pi c^3}\right)\frac{G^2{\rho }_1{\rho }_2}{a^2},\end{array}$$

(61)

whose sign corresponds to a repulsive interaction.

5 Conclusions

The goal of this review has been to summarize the modern picture of quantum gravity, within which the perturbative non-renormalizability of general relativity is recognized as being a particular instance of a more general phenomena: the widespread application of non-renormalizable quantum field theories throughout many branches of physics. Regarding quantum gravity in this way shows how quantitative predictions can be made: One must simply apply the rules of effective field theories, which are known to give an accurate description of experiments in low-energy nuclear, particle, atomic, and condensed matter physics.

Thinking of general relativity as an effective theory in this way is not a new development, and underlies most approaches to quantum gravity either explicitly or implicitly. Neither is it new to calculate explicitly the behaviour of quantum fields in curved space (sometimes including the graviton). What is new (over the last few years) is proceeding beyond the qualitative statement that general relativity is an effective theory to obtain the quantitative next-to-leading predictions within a controlled semiclassical approximation. Although much of the mechanics of such calculations leans on experience obtained when calculating with quantum fields in curved space, the crucial new difference is the quantitative power-counting arguments which identify precisely which quantum effects contribute to any given order in small quantities.

What emerges from this summary is a snapshot of a work which is very much still in progress.

The following loom large among the missing results:

• ∙ Although a general statement of the power-counting result for very light, relativistic particles near flat space has been known for some time [60, 27] , the central general power-counting results are not yet demonstrated to all orders for the most interesting case for practical calculations: the gravitational interactions of very massive, non-relativistic sources which are weakly interacting gravitationally (i.e., in spacetimes which are perturbatively close to Minkowski space).
• ∙ Although the theoretical tools exist, in the form of the operator product expansion, similarly general power-counting arguments are not yet given for quantum fluctuations about more general curved spaces.
• ∙ Some explicit calculations which are cast within the power-counting framework have been done, but many more can surely be done.

There can be little doubt that quantum effects are extremely small in the classical systems for which gravitational measurements are possible (like the solar system), but this need not undermine the motivation for their computation. The point of such calculations is not their relevance for practical experiments (we wish!). Rather, their point is conceptual. It is only through the careful calculation of quantum effects that the theory of their size can be solidly established. In particular, any precise comparison between observations and the predictions of classical gravity is ultimately incomplete unless the quantitative size of the quantum corrections is explicitly established, as a systematic, all-orders power-counting argument would do.

Furthermore, we can always hope to get lucky, even if only theoretically. A clean understanding of how the size of quantum corrections depends on the variables (mass, size, separation, etc.) in a given system, one might hope to find larger-than-generic quantum phenomena in special systems.

Even if these lie beyond the reach of present-day experimenters, they may furnish instructive theoretical laboratories within which differing approaches to quantum gravity might be more starkly compared.

In the last analysis, I hope the reader has become convinced of the utility of effective field theory techniques, and that the effective field theory point of view lifts the experimental triumphs of classical general relativity to precision tests of the leading-order implications of the quantum theory of gravity.

6 Acknowledgements

I wish to thank the editors of Living Reviews for the invitation to write this article, and for their great patience in waiting for its delivery. I am also grateful to John Donoghue and Ted Jacobson for their many useful comments on an early draft of this review. My own ideas on this topic are heavily indebted to Steven Weinberg, as conveyed through his graduate lectures on quantum field theory (and now by his textbooks on the subject).

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